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GIS Analysis Describing Attributes Statistical Analysis Spatial Description Spatial Analysis Searching for Spatial Relationships GIS and Spatial Analysis Introduction to GIS: Lecture #7 (GIS Analysis) Dueker (1979) • “a geographic information system is a special case of information systems where the database consists of observations on spatially distributed features, activities or events, which are definable in space as points, lines, or areas. A geographic information system manipulates data about these points, lines, and areas to retrieve data for ad hoc queries and analyses". Introduction to GIS: Lecture #7 (GIS Analysis) GIS is capable of data analysis • Attribute Data – Describe with statistics – Analyze with hypothesis testing • Spatial Data – Describe with maps – Analyze with spatial analysis Introduction to GIS: Lecture #7 (GIS Analysis) Attribute Description • The extremes of an attribute are the highest and lowest values, and the range is the difference between them in the units of the attribute. • A histogram is a two-dimensional plot of attribute values grouped by magnitude and the frequency of records in that group, shown as a variablelength bar. • For a large number of records with random errors in their measurement, the histogram resembles a bell curve and is symmetrical about the mean. Introduction to GIS: Lecture #7 (GIS Analysis) If the records are: • Text – Semantics of text e.g. “Hampton” – word frequency e.g. “Creek”, “Kill” – address matching • Example: Display all places called “State Street” Introduction to GIS: Lecture #7 (GIS Analysis) If the records are: • Classes – histogram by class – numbers in class – contiguity description, e.g. average neighbor (roads, commercial) Introduction to GIS: Lecture #7 (GIS Analysis) Describing a classed raster grid 20 P (blue) = 19/48 15 10 5 Introduction to GIS: Lecture #7 (GIS Analysis) If the records are: • Numbers – statistical description – min, max, range – variance – standard deviation Introduction to GIS: Lecture #7 (GIS Analysis) Statistical description • Range : min, max, max-min • Central tendency : mode, median (odd, even), mean • Variation : variance, standard deviation Introduction to GIS: Lecture #7 (GIS Analysis) Statistical description • Range : outliers • mode, median, mean • Variation : variance, standard deviation Introduction to GIS: Lecture #7 (GIS Analysis) Elevation (book example) Introduction to GIS: Lecture #7 (GIS Analysis) GPS Example Data: Elevation Table 6.2: Sample GPS Readings Data Extreme Date Time D M S Minimum Maximum Range 6/14/95 6/15/95 1 Day D MS 10:47am 42 30 54.8 75 41 13.8 10:47pm 42 31 03.3 75 41 20.0 12 hours 00 8.5 00 6.2 Introduction to GIS: Lecture #7 (GIS Analysis) Elev 247 610 363 Mean • Statistical average • Sum of the values for one attribute divided by the number of records n X = X i /n i = 1 Introduction to GIS: Lecture #7 (GIS Analysis) Computing the Mean Sum of attribute values across all records, divided by the number of records. Add all attribute values down a column, / by # records A representative value, and for measurements with normally distributed error, converges on the true reading. A value lacking sufficient data for computation is called a missing value. Does not get included in sum or n. Introduction to GIS: Lecture #7 (GIS Analysis) Variance The total variance is the sum of each record with its mean subtracted and then multiplied by itself. The standard deviation is the square root of the variance divided by the number of records less one. For two values, there is only one variance. Introduction to GIS: Lecture #7 (GIS Analysis) Standard Deviation Average difference from the mean st.dev. Sum of the mean subtracted from the value for each record, squared, divided by the number of records1, square rooted. = (X i - X ) n-1 Introduction to GIS: Lecture #7 (GIS Analysis) 2 GPS Example Data: Elevation Standard deviation Same units as the values of the records, in this case meters. Average amount readings differ from the average Can be above of below the mean Elevation is the mean (459.2 meters) plus or minus the expected error of 82.92 meters Elevation is most likely to lie between 376.28 meters and 542.12 meters. These limits are called the error band or margin of error. Introduction to GIS: Lecture #7 (GIS Analysis) The Bell Curve Mean 12.17 % 484.5 459.2 37.83 % Introduction to GIS: Lecture #7 (GIS Analysis) Samples and populations • A sample is a set of measurements taken from a larger group or population. • Sample means and variances can serve as estimates for their populations. • Easier to measure with samples, then draw conclusions about entire population. Introduction to GIS: Lecture #7 (GIS Analysis) Testing Means Mean elevation of 459.2 meters standard deviation 82.92 meters what is the chance of a GPS reading of 484.5 meters? 484.5 is 25.3 meters above the mean 0.31 standard deviations ( Z-score) 0.1217 of the curve lies between the mean and this value 0.3783 beyond it Introduction to GIS: Lecture #7 (GIS Analysis) Hypothesis testing • Set up NULL hypothesis (e.g. Values or Means are the same) as H0 • Set up ALTERNATIVE hypothesis. H1 • Test hypothesis. Try to reject NULL. • If null hypothesis is rejected alternative is accepted with a calculable level of confidence. Introduction to GIS: Lecture #7 (GIS Analysis) Testing the Mean • Mathematical version of the normal distribution can be used to compute probabilities associated with measurements with known means and standard deviations. • A test of means can establish whether two samples from a population are different from each other, or whether the different measures they have are the result of random variation. Introduction to GIS: Lecture #7 (GIS Analysis) Alternative attribute histograms Skewed above mean No error in measurement All measurements equally right Random with some systematic error Decrease away from zero Bimodal (two averages) Introduction to GIS: Lecture #7 (GIS Analysis) Normal distribution Random distribution Accuracy Determined by testing measurements against an independent source of higher fidelity and reliability. Must pay attention to units and significant digits. Can be expressed as a number using statistics (e.g. expected error). Accuracy measures imply accuracy users. Introduction to GIS: Lecture #7 (GIS Analysis) The difference is the map • GIS data description answers the question: Where? • GIS data analysis answers the question: Why is it there? • GIS data description is different from statistics because the results can be placed onto a map for visual analysis. Introduction to GIS: Lecture #7 (GIS Analysis) Spatial Statistical Description • For coordinates, the means and standard deviations correspond to the mean center and the standard distance • A centroid is any point chosen to represent a higher dimension geographic feature, of which the mean center is only one choice. • The standard distance for a set of point spatial measurements is the expected spatial error. Introduction to GIS: Lecture #7 (GIS Analysis) Spatial Statistical Description • For coordinates, data extremes define the two corners of a bounding rectangle. Introduction to GIS: Lecture #7 (GIS Analysis) Mean Center mean y mean x Introduction to GIS: Lecture #7 (GIS Analysis) Centroid: mean center of a feature Introduction to GIS: Lecture #7 (GIS Analysis) Comparing spatial means Introduction to GIS: Lecture #7 (GIS Analysis) GIS and Spatial Analysis Descriptions of geographic properties such as shape, pattern, and distribution are often verbal Quantitative measure can be devised, although few are computed by GIS. GIS statistical computations are most often done using retrieval options such as buffer and spread. Also by manipulating attributes with arithmetic commands (map algebra). Introduction to GIS: Lecture #7 (GIS Analysis) Statistics and features HIGH VALUE MEDIUM VALUE LOW VALUE Points Bounding rectangle Standard distance Lines Number of points Length of line Length index Areas Area in square units Boundary length Number of Holes Area/area of bounding rectangle Area of largest enclosed circle Figure 6.8 Numbers that describe points, lines, and areas. Introduction to GIS: Lecture #7 (GIS Analysis) An example • • • • • Lower 48 United States 1996 Data from the U.S. Census on gender Gender Ratio = # females per 100 males Range is 96.4 - 114.4 What does the spatial distribution look like? Introduction to GIS: Lecture #7 (GIS Analysis) Gender Ratio by State: 1996 Introduction to GIS: Lecture #7 (GIS Analysis) Searching for Spatial Pattern A linear relationship is a predictable straightline link between the values of a dependent and an independent variable. (y = a + bx) It is a simple model of the relationship. A linear relation can be tested for goodness of fit with least squares methods. The coefficient of determination r-squared is a measure of the degree of fit, and the amount of variance explained. Introduction to GIS: Lecture #7 (GIS Analysis) Simple linear relationship best fit regression line y = a + bx observation dependent variable gradient intercept y=a+bx independent variable Introduction to GIS: Lecture #7 (GIS Analysis) Testing the relationship gr = 117.46 + 0.138 long. Introduction to GIS: Lecture #7 (GIS Analysis) Patterns in Residual Mapping Differences between observed values of the dependent variable and those predicted by a model are called residuals. A GIS allows residuals to be mapped and examined for spatial patterns. A model helps explanation and prediction after the GIS analysis. A model should be simple, should explain what it represents, and should be examined in the limits before use. We should always examine the limits of the model’s applicability (e.g. Does the regression apply to Europe?) Introduction to GIS: Lecture #7 (GIS Analysis) Mapping residuals from a model Introduction to GIS: Lecture #7 (GIS Analysis) Unexplained variance • • • • • • • More variables? Different extent? More records? More spatial dimensions? More complexity? Another model? Another approach? Introduction to GIS: Lecture #7 (GIS Analysis) Example: Spatial Analysis 12 14 19 10 40 12 6 25 ? 14 11 30 meters to water table resolution? extent? accuracy? precision? Introduction to GIS: Lecture #7 (GIS Analysis) boundary effects? point spacing? GIS and Spatial Analysis Geographic inquiry examines the relationships between geographic features collectively to help describe and understand the real-world phenomena that the map represents. Spatial analysis compares maps, investigates variation over space, and predicts future or unknown maps. Many GIS systems have to be coaxed to generate a full set of spatial statistics. Introduction to GIS: Lecture #7 (GIS Analysis) Analytic Tools and GIS Tools for searching out spatial relationships and for modeling are only lately being integrated into GIS. Statistical and spatial analytical tools are also only now being integrated into GIS, and many people use separate software systems outside the GIS. Real geographic phenomena are dynamic, but GISs have been mostly static. Time-slice and animation methods can help in visualizing and analyzing spatial trends. GIS places real-world data into an organizational framework that allows numerical description and allows the analyst to model, analyze, and predict with both the map and the attribute data. Introduction to GIS: Lecture #7 (GIS Analysis)